We have studied the phase behaviour of two different kinds of molten heteropoly-mers with quenched randomness: random block copolymer melts and crosslinked homopolymer blends. In both systems, the frustration between the enthalpy of mixing and entropic forces hampers or prevents macroscopic phase separation, giving rise to microstructured phases with a variety of possible morphologies.
In chapter 2 we have addressed random block copolymers using different ap-proaches derived from the same microscopic model. By coarse-graining the model, disregarding the conformations of the individual chains, we have recovered the mul-ticomponent picture by Nesarikar et al. [16], which divides the copolymer chains into species according to their content of A and B monomers and determines the chemical equilibrium separately for each species. Due to the coarse-graining, the theory is restricted to separations into macroscopic phases, for which it is exact on the mean-field level also in the case of first order transitions. The analytic equilibrium conditions have been solved numerically. The alternative approach by Fredrickson et al. [17], based on a fourth-order Landau expansion of the effective free energy of the system, retains the microscopical details of the model and is thus capable of describing the separation into microscopic phases. We have extended the theory by Fredricksonet al.in different ways. We have shown that certain ap-proximations for long blocks and chains in ref. [17] are inconsistent. For instance, they lead to the incorrect predictions that the system is insensitive to the mi-crophase morphology and that the mimi-crophases reveal an infinite domain spacing at their onset in the symmetric case. According to our calculation, microphases in a symmetric melt always set in with finite wavelength and lamellar morphology.
Abstaining from using said approximations, we have also been able to consider the case of short blocks and short chains, finding considerable deviations from the long-chain predictions. Furthermore, we have discussed the effects of compressibil-ity, which faciliates phase separation and leads to a higher spinodal temperature in the asymmetric case. Comparing the predictions of the Landau theory and the multicomponent theory for macroscopic phase separation shows that the Landau expansion is only correct for nearly symmetric melts. For asymmetric melts, the transition becomes first-order and the Landau expansion breaks down.
Monte-Carlo simulations by Houdayer and M¨uller [21] suggest a fractionation of the melt according to the chain sequences, leading to the coexistence macro- and microphases, with homopolymeric chains preferring the former and copolymeric chains preferring the latter. In order to study the coexistence of homogeneous and microstructured phases, and, in particular, fractionation as a mechanism enabling microphase separation, we have set up a simple fractionation scheme. We have
shown that fractionation is sufficient to enable microphase separation, without requiring the subtle relation between the spatial dependence of the second and fourth-order vertices of the Landau expansion necessary in the conventional theory by Fredrickson et al. [17]. Moreover, unlike ref. [22], our approach allows for the direct observation of the partitioning of the chains onto the three phases according to their sequence.
In chapter 3 we have developed a comprehensive model for randomly cross-linked homopolymer blends describing both gelation and phase separation in a microscopic fashion free of ad hoc assumptions. Moreover, our model accounts for different external conditions prior to and after preparation. In agreement with ex-periments [49] and previous phenomenological theories [45, 52, 53], we have found the stability of the mixed state to be enhanced by the introduction of crosslinks.
We have computed the degree of stabilisation quantitatively for symmetric and asymmetric blends. Furthermore, our theory allowed us to calculate the scatter-ing functions describscatter-ing volatile and glassy, i.e. persistent, demixing fluctuations.
Independently varying the conditions in the preparation and measurement ensem-bles and the amount of crosslinking, we have identified three regimes, in which the glassy fluctuations are dominated by either the pre-crosslinking fluctuations that are frozen-in into the gel network, by fluctuations seeded by frozen-in disorder, or by the fluctuations towards microphase separation. The latter are peaked at wavenumbers of about the inverse localisation length of the gel; they announce, and diverge at, the microphase transition.
The microphase separated state itself has been investigated considering different morphologies, including a random pattern. In incompressible symmetric systems, the lowest free energy is achieved by a lamellar lattice with a wavelength that corresponds to the localisation length of the gel and that only slightly varies with temperature on going deeper into the phase separated regime. Asymmetric com-positions lead to a first-order transition, and the favoured microphase morphology are spherical domains of the minority component on a bcc lattice the matrix of the majority component in this case. Although mostly focusing on the incompressible limit, we have also studied the effects of compressibility, which partly contribute even in the incompressible limit. Moreover, we have shown that sufficiently com-pressible system prefer a random morphology to the regular microphase patterns.
Further steps
To generalise our theory of fractionation discussed in section 2.7 to the case of longer chains, it would be helpful to extend the class of polymer sequences allowed to “choose” between micro- and macrophases. Moreover, we have concentrated on fractionation of symmetric systems so far; it would be interesting to consider the asymmetric case, as well. Further points worth taking into account are the effects of fluctuations and polydispersity of blocks and chains. Since the enthalpy of mixing increases with the length of blocks and chains, whereas the entropy of mixing per chain is fixed, polydispersity has a major influence on the phase
the chain length: the longest chains separate first to form microstructured phases, which coexist with the disordered phase of smaller chains [31].
In the discussion of crosslinked homopolymer blends in chapter 3, we concen-trated, for the sake of simplicity, on the case of chains with equal degree of poly-merisation before crosslinking. It would be interesting to disengage from this idealisation and to also study blends in which the two chain types have different lengths (still being monodisperse within the species); this should be technically straightforward. (The consideration of polydispersity would probably require ma-jor modifications of our theory.) A further possible extension is to consider a probability of crosslinking that depends on the monomer species involved. Such a model would cover, as limiting cases, interpenetrating and semi-interpenetrating networks, in which only chains of the same type or even chains of only one of the two types are crosslinked, respectively. It might also appear natural to go beyond mean-field theory, although, as pointed out before, mean-field theory is largely valid in the case of homopolymer blends [64].
Our technique appears well-suited to consider other crosslinked systems of in-teracting particles, as well. In our group, there are currently activities on nematic elastomers, with the long-term goal of developing a theory for spider silks [74].